A Framework for Parallel PDE Solvers on Multiscale Adaptive Cartesian Grids

In many fields of application in science and engineering, the grid-based numerical simulation of partial differential equations leads to new scientific insights due to increasing computing resources, increasing amounts of data, and increasing efficiency of the algorithms used. All three of them facilitate more detailed models and more reliable simulations. Yet, a growing code complexity accompanies this progress. To tackle this complexity, more and more solvers for partial differential equations rely on frameworks. State-of-the-art frameworks have to support multiscale algorithms for arbitrary dimensional problems with dynamic, i.e. changing, discretisations. Despite this flexibility of data and data access, the realisation has to have low memory requirements, as the gap between computing power and memory bandwidth and access speed broadens. It furthermore has to exploit modern computer architectures and has to scale on parallel computers, even if the data structures and the data access pattern change permanently. This thesis presents a framework tackling these challenges with adaptive Cartesian grids resulting from a generalised octree concept. They are traversed with space-filling curves and a small yet fixed number of stacks acting as data containers. Within this context, dynamically changing multiscale grids of arbitrary dimension can be stored with a few bits per datum, whereas classical approaches often require several kilobytes and more flexible data structures entailing a runtime overhead. The thesis formalises the approach and reduces the implementation complexity—the algorithmic principle itself has been well-known for several years—from an exponential to a linear number of containers in the spatial dimension of the problem. A modification of the grid traversal originally following a depth-first order then facilitates a domain decomposition strategy with dynamic load-balancing. The framework is named after the Italian mathematician Giuseppe Peano who discovered the underlying space-filling curve. Its potential is demonstrated via a geometric multigrid solver for the Poisson equation with low memory requirements, good cache hit rates, a posteriori adaptive grids, and a dynamic load balancing— a combination of characteristics rarely found. As the characteristics result from the framework usage, the thesis paves the way to a multiscale computational fluid dynamics application on instationary, hierarchical grids that uses Peano.